Base point freeness conjecture for nef and big Kähler classes

Let $(X,B)$ be a klt pair, where $X$ is a normal $Q$-factorial compact Kähler variety. Let $\alpha\in H^{1,1}_{\rm BC}(X)$ be a nef class on $X$. If $\alpha-(K_X+B)$ is nef and big, then the base point freeness conjecture. There exists a proper surjective morphism with connected fibers

$$f:X\to Y$$

to a normal compact Kähler variety $Y$ with rational singularities and a Kähler class $\alpha_Y\in H^{1,1}_{\rm BC}(Y)$ such that

$$\alpha=f^*\alpha_Y.$$

This is the Kähler analogue of the projective base-point-free theorem, formulated for nef and big cohomology classes. The source presents it as a conjecture for Kähler varieties; the supplied material gives no evidence that it has been resolved.